POISSON'S PROBABILITY SUMMATION 613 



resenting the corresponding l\)isson distribution terminated at 

 P = l A' and P=l — 1N, and with the observed points in the range 

 P=10 N to P=l—10N shown as soHd black dots and the points 

 outside this range shown as circles with white centers. This sec- 

 ondar\- division is quite arbitrary, for the increase in relial)ility of 

 the points as the center of the range is approached is gradual. There 

 will, of course, be irregularities due to sampling even in the center as 

 long as the number of samples is finite. 



Non-uniforniit\- of the samples of the series may introduce a definite 

 trend away from the Poisson distribution, a slant to the right such 

 as is shown in Figs. 7 and 8. Such trends result when the value of a 

 varies from sample to sample of the series. Fig. 7 shows three theo- 

 retical distributions of this sort, each having the same average a = 75. 

 Series (a) is made up of two equal sub-series having a = 50 and a = 100, 

 respectively, (b) of two unequal sub-series, in the ratio of 3:1, having 

 a = 60 and a = 120, respectively, and (c) of three equal sub-series having 

 a = 15, a = 60, and a = 150, respectively.^ Fig. 8 shows the effect on 

 the distribution of letting a vary continuously and uniformly between 

 the limits 5 and 15, the compound series (b) made up of two equal 

 sub-series with averages 5 and 15 being also shown for comparison.'^ 

 Since in practical time series a usually increases or decreases with the 

 time, this kind of distribution may be expected to occur frequently. 

 It should be noted that in all these cases it is immaterial whether a 

 changes because of a change in the number of trials in the sample, or 

 because of a change in the probability of the event's happening at a 

 single trial, or because of both; if a is constant throughout the series a 

 Poisson distribution will be obtained, and if a varies the tendency to 

 slope to the right will be introduced. Various devices may be em- 

 ployed to keep the average constant in an actual series, some of 

 which will be illustrated by the examples given below. 



In selecting the following examples of the Poisson summation only 

 two general rules were followed : that there must be some reason to 



^ In a compound distribution 



p=s^p. 



where iV,- is the number of samples with the average a,-, and Pi = P{c, a,). 



^° If a varies uniformly and continuously from ai to az 

 fas P(c, a) 



(02 



02 — ai 



da 



= 1 ~ i:iP{i,a2)-Pii,a,)-\. 



02 — ai 



•=u 



